RDP del Pezzo surfaces with global vector fields in odd characteristic

TL;DR

This paper classifies RDP del Pezzo surfaces with global vector fields over algebraically closed fields of various characteristics, revealing new non-equivariant examples in characteristics 3, 5, and 7, and constructing explicit surfaces over imperfect fields.

Contribution

It extends the classification of RDP del Pezzo surfaces with global vector fields to positive characteristics, especially identifying non-equivariant cases in characteristics 3, 5, and 7.

Findings

Classified RDP del Pezzo surfaces with global vector fields in characteristic p ≠ 2.

Identified non-equivariant RDP del Pezzo surfaces in characteristics 3, 5, and 7.

Constructed explicit examples over imperfect fields in characteristic 7.

Abstract

We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic $p \neq 2$. In characteristic $0$, every RDP del Pezzo surface $X$ is equivariant, that is, ${\rm Aut}_X = {\rm Aut}_{\widetilde{X}}$, where $\widetilde{X}$ is the minimal resolution of $X$, hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if $p \neq 2,3,5,7$, then it is still true that every RDP del Pezzo surface is equivariant. We classify the non-equivariant RDP del Pezzo surfaces in characteristic $p = 3,5,7$, giving explicit equations for every such RDP del Pezzo surface in all possible degrees. As an application, we construct regular non-smooth RDP del Pezzo surfaces over imperfect fields of characteristic $7$, thereby showing that the known bound $p \leq 7$ for the characteristics, where such a surface can exist, is sharp.